13 ms·

After centuries, a seemingly simple math problem gets an exact solution

305 points by techlover14159 6y ago

Invictus0 6y ago
Plaudits to Steve Nadis for pulling every goat joke known to man for this article.
bryan0 6y ago
> Unfortunately, there’s a catch. Ullisch’s solution ... can’t tell you, in a practical sense, how long to make the goat’s leash. Approximations are still required... Ok, then in what sense is this an exact solution?
- whatshisface 6y ago
  In the sense that sqrt(2) can be an exact solution, even though you can't write it down as a decimal.
  - gowld 6y ago
    No, sqrt(2) is exact and no approximation is needed -- it's the diagnoal of a 1x1 square. Goats and grass don't care about decimal number systems. As the article says, the solution to the goat problem isn't practically constructible: "It’s a bit more abstruse — the ratio of two so-called contour integral expressions, with numerous trigonometric terms thrown into the mix"
    - whatshisface 6y ago
      >No, sqrt(2) is exact and no approximation is needed -- it's the diagonal of a 1x1 square. That's like saying that the goat-grass system as described is exact, and no approximation is needed. I can write 1 x sqrt(2) just as easily as I can write 1 x (goat-grass constant). We arbitrarily choose what constants and symbols are allowed when we use the phrase "exact solution." Philosophically, every solution is at most breaking down an answer into other solutions.
    - Koshkin 6y ago
      Problem is, if you make sqrt(2) exact - say, by an appropriate choice of the number system - then 1 will not be exact. This is because the ratio sqrt(2) / 1 is irrational in any number system.
      - thaumasiotes 6y ago
        But this isn't true. As gowld points out, you can easily represent an exact 1 and an exact sqrt(2) simultaneously; all you need is compass and straightedge. Interestingly, that's as far as it goes; if you start with a line of length 1, compass and straightedge will let you construct a line with length equal to the square root of any rational number. You can do addition, multiplication, subtraction, division... and square roots, and there things come to a stop.
        whatshisface 6y ago
        But what singles out a compass and straightedge as a "the" construction system? There are infinity other ways to construct numbers, each one with a potentially different set of constructable numbers.
        thaumasiotes 6y ago
        You can use them to draw lines which you can then use to cut leather. They don't need to be "the" construction system; they are a construction system, and a very simple one.
        whatshisface 6y ago
        A pencil tied to the end of a string which is itself tied to the outside of a circle is also a construction system, one that happens to be able to construct the goat-grass area exactly. :)
        thaumasiotes 6y ago
        > one that happens to be able to construct the goat-grass area exactly. How? How did you get a string of the correct length? Note that the tool you described is a compass.
  - deleted 6y ago
    [deleted]
- castratikron 6y ago
  It's in closed form
- svat 6y ago
  It's clearer from the Wikipedia article (https://en.wikipedia.org/w/index.php?title=Goat_problem&oldid=993368252 https://en.wikipedia.org/w/index.php?title=Goat_problem&oldi...): earlier we only had expressions for r like https://wikimedia.org/api/rest_v1/media/math/render/svg/9758becd77e7b4102ccdf50ff0845a749d236dc2 https://wikimedia.org/api/rest_v1/media/math/render/svg/9758... and https://wikimedia.org/api/rest_v1/media/math/render/svg/a63430b98b5e138a3d73831a2b7ecb6825be7a5b https://wikimedia.org/api/rest_v1/media/math/render/svg/a634... (from which of course one could numerically compute the answer r=1.1587…), but now thanks to this paper we have the expression https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7451ea92f107e56651328ad15bfcc3276f87f2 https://wikimedia.org/api/rest_v1/media/math/render/svg/5b74... that does not involve r on the right-hand side.
  - Sniffnoy 6y ago
    Hm, if it's based on this ratio of contour integrals, shouldn't it be possible to do better than this? Like why would it be so hard to find the residues for these poles? Shouldn't that be just a bit of formal Laurent series manipulation? What am I missing here?
    - Someone 6y ago
      If the series don’t cancel out nicely (likely, I would guess), wouldn’t you end up with some infinite sum? If so, as this example shows (contour integrals in a closed form?) “closed form” is loosely defined, but I think most would say something with an infinite sum wouldn’t be one (but then, https://en.wikipedia.org/wiki/Closed-form_expression#Analytic_expression https://en.wikipedia.org/wiki/Closed-form_expression#Analyti... says: “In particular, special functions such as the Bessel functions and the gamma function are usually allowed, and often so are infinite series and continued fractions. On the other hand, limits in general, and integrals in particular, are typically excluded.[citation needed]”
      - Sniffnoy 6y ago
        No, you shouldn't end up with an infinite sum; if you only want to know the series finitely far out, you only need to know finitely many terms. However, it seems the problem is hard for other reasons. I had assumed, without checking, that they'd put the center of the circle at 3pi/8 because it's a pole. Nope! As best I can tell from some graphing tools, there is indeed precisely one pole in the circle, and it's on the real line, but it's close to the right endpoint; I don't know that it has any nice form. So I imagine that getting any sort of exact series expansion around there -- or even just getting exactly the first few terms, i.e. the first few derivatives there, which would be all you'd need -- would be difficult for that reason. (Although, the higher the order of the zero, the more initial terms you'd need...) Edit: Actually, I guess it looks like a zero of order 1? Except that doesn't make sense, because then the top contour integral would be zero...
- 6gvONxR4sf7o 6y ago
  Imagine an exact solution like this: sin(cos(log(98234/123)+tan(exp(pi/5))-1)+pi + integral(complicated function(x) dx from 0 to 1) It doesn't tell you in a practical sense how long to make the goat's leash. So you find the numerical version to however many decimals you want to get a useful approximation.
- pfortuny 6y ago
  The unknown is isolated: r=very-difficult-to-compute-expression
- elil17 6y ago
  The word exact here is an inexact description of what sort of solution this really is - it's a closed form explicit solution. A closed form solution means that the equation is limited to to certain common mathematical operations and is finite in length. An explicit solution means that the quantity we are solving for is isolated on one side of the equation.
2sk21 6y ago
Is this in any way to the problem of quadrature of the lune? https://en.wikipedia.org/wiki/Lune_of_Hippocrates https://en.wikipedia.org/wiki/Lune_of_Hippocrates
svat 6y ago
The paper this article is about: * A Closed-Form Solution to the Geometric Goat Problem, by Ingo Ullisch in The Mathematical Intelligencer. https://doi.org/10.1007/s00283-020-09966-0 https://doi.org/10.1007/s00283-020-09966-0 The illustration in the article (I mean the top half of https://d2r55xnwy6nx47.cloudfront.net/uploads/2020/12/Grazing-goat-figure.svg https://d2r55xnwy6nx47.cloudfront.net/uploads/2020/12/Grazin...) was really helpful to understand what the question is. I know Quanta Magazine articles sometimes get some negative comments here on HN from readers who expect a different kind of exposition than what is suitable for a magazine of that sort, but for my part I'm really grateful to Quanta Magazine for bringing attention to papers like this, and writing nice articles about them with history, good illustrations, quotes from other mathematicians, etc. For the impatient, Wikipedia has a short article on the problem and earlier progress: https://en.wikipedia.org/w/index.php?title=Goat_problem&oldid=993368252 https://en.wikipedia.org/w/index.php?title=Goat_problem&oldi... From this it's more clear in what sense this is progress: whereas earlier the answer r=1.1587284730181215178... (https://oeis.org/A133731 https://oeis.org/A133731) was known via more than one formula of the form r = (some function of r), only now do we have a closed-form expression of the form r = (some expression not involving r).
tunesmith 6y ago
I first misread "goat tied to the inside of the fence" as meaning tied to a post in the center of the circle, and was confused at why everyone thought this question was so hard.
- mrlala 6y ago
  I was just drawing what you said on the whiteboard, and I was like.. uh, am I missing something or am I a super genius? I am positive the latter is incorrect. I had to re-read the first paragraph multiple times until I understood the goat was not tethered to the center.
  - InitialLastName 6y ago
    There's a picture of the problem like halfway down the article.
    - mrlala 6y ago
      Yeah but I wanted to attempt to think about the problem myself before reading more on the article in case they gave away a very simple clue.
- fastaguy88 6y ago
  I initially assumed that the goat was tied to the fence in such a way that the non-goat rope-end could move completely around the circumference of the fence and not be fixed to a single point.
  - acqq 6y ago
    I was also at first confused, and I find the problem as published in 1894 (i.e. almost 130 years ago) much clearer: "A circle containing one acre is cut by another whose centre is on the circumference of the given circle, and the area common to both is one-half acre. Find that radius of the cutting circle." I think that formulation had to be at least mentioned near the beginning of the text, when not used in the first sentence.
zaptheimpaler 6y ago
Finally! This was getting to be a real problem in my grazing goat as a service business.
- hinkley 6y ago
  Goats as a Service has been around longer than The Cloud. I think you already missed that gold rush.
- _jal 6y ago
  You laugh, but growing up, my family used goats to keep brush down on our property, and would occasionally loan them to neighbors for the same.
  - mhh__ 6y ago
    https://www.telegraph.co.uk/technology/google/5297097/Google-hires-goats-to-cut-grass.html https://www.telegraph.co.uk/technology/google/5297097/Google...
- skinkestek 6y ago
  I think that is an actual service provided in Norway.
  - codetrotter 6y ago
    You are correct. I remember seeing some goats at Oscarsborg Fortress and I was told that they were there on loan or on hire. Looked it up now and it checks out. The articles in the two links below affectionately refer to the featured goats as the "coast goat commando", which I think is just lovely :) https://www.oblad.no/badebyen/kystgeitkommandoen-til-oscarsborg/s/2-2.2610-1.3865150 https://www.oblad.no/badebyen/kystgeitkommandoen-til-oscarsb... https://www.oblad.no/badebyen/umb-geitene-pa-sommerjobb/s/2-2.2610-1.3884117 https://www.oblad.no/badebyen/umb-geitene-pa-sommerjobb/s/2-... Both of these articles are in Norwegian but basically they talk about the importance of keeping the vegetation in check, and that the goats are great at this, as well as the social value that goats provide to visitors. The goats featured here are owned by a University and were rented out to the Oscarsborg Fortress. There are pictures of the goats also in the articles. PS: For anyone not familiar with Oscarsborg check out the following links for some info and pictures: https://en.wikipedia.org/wiki/Oscarsborg_Fortress https://en.wikipedia.org/wiki/Oscarsborg_Fortress https://www.forsvarsbygg.no/no/festningene/finn-din-festning/oscarsborg-festning/ https://www.forsvarsbygg.no/no/festningene/finn-din-festning... https://en.wikipedia.org/wiki/Battle_of_Drøbak_Sound https://en.wikipedia.org/wiki/Battle_of_Drøbak_Sound Using the defence batteries at Oscarsborg Fortress, the Norwegian coastal defence successfully sank the German heavy cruiser Blücher on 9 April 1940, forcing the German fleet to fall back. https://en.wikipedia.org/wiki/German_cruiser_Blücher https://en.wikipedia.org/wiki/German_cruiser_Blücher
atsmyles 6y ago
Hey Terence Tao, you might be a great mathematician but Ingo Ullisch is the GOAT!
deleted 6y ago
[deleted]
nullc 6y ago
Why do I have to go to Wikipedia to see the solution? Would it have been that hard to stick https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7451ea92f107e56651328ad15bfcc3276f87f2 https://wikimedia.org/api/rest_v1/media/math/render/svg/5b74... in the article?
- ABeeSea 6y ago
  From a journalistic standpoint, including the formula would require a longer digression into contour integrals than the one clause the article currently contains. It’s also not clear what license that image is available under.
  - segfaultbuserr 6y ago
    > It’s also not clear what license that image is available under. A plain image of mathematical equation is not copyrightable, it's literally the MathJax output of a LaTeX equation ({\displaystyle r=2\cos \left({\frac {1}{2}}{\frac {\oint _{|z-3\pi /8|=\pi /4}z/(\sin z-z\cos z-\pi /2)\,dz}{\oint _{|z-3\pi /8|=\pi /4}1/(\sin z-z\cos z-\pi /2)\,dz}}\right)}). It does not have any copyrightable artistic design. And to the nitpickers - the pixmap output of a font is also not copyrightable under U.S. copyright laws. Even if it is, Computer Modern is available under a free license. And even if it's not, pure facts - such as math formulas - are not copyrightable, it would be trivial to write down the identical equation using another program and font. Copyright is not an inevitable, divine, or natural right, it is only an utilitarian tool adopted by the Constitution and lawmakers "to promote the Progress of Science and useful Arts." Thus, there always exist things that cannot be copyrighted. It's also why fair use of copyrighted works is conditionally allowed (not relevant to this case). The tendency of people to assume that every single piece of data is automatically controlled exclusively under copyright is frustrating.
  - blu_ 6y ago
    > Content is available under CC BY-SA 3.0 unless otherwise noted Just including the actual solution in an article about this problem would be IMO a much better choice.
- 6y ago
crazygringo 6y ago
It's fascinating how a problem that looks at first glance like it belongs as a basic homework problem in a calculus textbook turns out to be so difficult. I'm curious if there are lists of other problems that are similarly easy to understand in a few seconds, that seem like they'd be similarly easy to solve, but which turn out to be fiendishly hard like this one? Especially ones that can be visualized easily geometrically like this one.
- darkhorse13 6y ago
  >I'm curious if there are lists of other problems that are similarly easy to understand in a few seconds I too would love such a list. Can anyone point me if one does exist?
  - deleted 6y ago
    [deleted]
- FartyMcFarter 6y ago
  I'm struggling to come up with examples. Maybe the 3n+1 problem? It's definitely easy to understand in a few seconds, and definitely fiendishly hard. I'm not sure if it seems easy to solve though :) Another one that came to mind is about efficiently computing the permanent of a matrix[1]. Maybe understandable in seconds if you know how determinants work. [1] https://en.wikipedia.org/wiki/Computing_the_permanent https://en.wikipedia.org/wiki/Computing_the_permanent
  - DataDaoDe 6y ago
    3n + 1 is interesting because it seems so tantalizingly easy and beautiful at first glance. Its like looking at a stream where you can clearly see the stony bed and it appears to be ankle deep but once you take a step you realize the lighting has fooled you and you fall head over heels into icy waters.
    - pas 6y ago
      It's so much more interesting if you happen to know about the "hydra game" (the Goodstein theorem), which is also a question about sequences and their termination at 0. https://en.wikipedia.org/wiki/Goodstein%27s_theorem https://en.wikipedia.org/wiki/Goodstein%27s_theorem But this is easily proved (using infinite ordinals, but it seems it could be proved just by coming up with a similar concept like the infinite ordinal arithmetic, basically providing an upper bound for each step of the algorithm and showing that there's an eventual maximum to these and then there's a monotonic decrease, and that the number of steps are always finite).
r-bryan 6y ago
Cool. Now solve it for a circular fence on the surface of a sphere. In fact, solve all four cases {exterior, interior of sphere} X {exterior, interior of fence}. Spherical trig can only make the solution(s) even more heroic, right?
- Akronymus 6y ago
  Why not try hyperbolic space too?
- antiquark 6y ago
  You forgot about the hypergoats.
- scatters 6y ago
  Surface of a sphere depends on how large the fence is compared to the sphere. If the fence is small, the answer is the same as the plane version. If the fence is as large as possible (an equator) then the rope needs to be precisely one quadrant in length, equalling the "radius" of the fence. If the fence encloses more than half the sphere... well, if it encloses all of the sphere (that is, it is a small circle with the "inside" declared to be the outside) then the rope is again one quadrant, so half the "radius" of the fence. More interesting is a space-goat tethered to the interior of a hollow sphere, hypersphere etc.; no closed-form solutions for higher-dimensional cases, but the answer tends to sqrt(2) as the number of dimensions approaches infinity.
cochne 6y ago
Not really an exact or closed form solution since it contains an integral. It is an explicit solution compared to the implicit one we had before.
- adwn 6y ago
  > Not really an exact [...] solution But it is exact. > Not really [a] closed form solution It's my understanding that what is and isn't "closed form" is rather arbitrary. Functions which are used frequently – like exp() – are elevated to closed form status, and yet you can't evaluate exp() in a finite number of steps. So how is the explicit solution to the goat problem objectively different?
  - cochne 6y ago
    Yes I also feel that way to some degree, but I just never considered an integral to be closed form. There is some argument to be made for exp, as it is considered an 'elementary function'. I was going off this statement from wikipedia on closed form expressions "It may contain constants, variables, certain "well-known" operations (e.g., + − × ÷), and functions (e.g., nth root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit, differentiation, or integration." Edit: I think if you say an integral is closed form, you must also admit that a limit is closed form, since an integral is defined in terms of limits (though technically more restrictive). In that case, you should also admit that we already had a closed form expression for this number, as it could be expressed as a limit of an iterative process.
    - adwn 6y ago
      > There is some argument to be made for exp, as it is considered an 'elementary function'. But exp() is defined as the limit over an infinite sum, so why does it get to be an elementary function? My point it that the distinction between closed form and non-closed form is arbitrary, and that there is no qualitative difference. In fact, limit and integral are just (higher-order) functions as well – and rather ubiquitous ones, so why aren't they considered elemental?
skinkestek 6y ago
This is probably written by people who don't have real-life experience with goats I guess. I cannot even concentrate on the problem since I keep coming back to what the goat will do: - jump the fence - chew the rope - probably more
- yetihehe 6y ago
  In such cases, it's best to assume it's a special zero dimensional point goat, which only chews grass.
  - adwn 6y ago
    ...in a vacuum.
  - cgriswald 6y ago
    How do you tie a zero dimensional point goat? How do you know he's in the fence at all? How does he chew the grass? Where does the grass go?
    - tuatoru 6y ago
      And having discovered the goat, how do you put a tether on something with no size?
    - WJW 6y ago
      You'd need a special rope to tie it as well, in addition to low-dimensional fencing and specialized grass.
    - thechao 6y ago
      Literally every one of these questions pertains to real goats, too.
    - morei 6y ago
      ... are you serious? You use an infinitely thin rope of course! Slightly more seriously, you take the limit of a alpha-sized goat tied with an alpha-sized rope as alpha goes to zero from above.
tuatoru 6y ago
From TFA: * Of course, it won’t upend textbooks or revolutionize math research, Ullisch concedes, because this problem is an isolated one. “It’s not connected to other problems or embedded within a mathematical theory.”* A island peak hinting at a submerged continent of mathematics. Unfortunately since our brains evolved (under a regime of calorie cost vs survival benefit) and are therefore limited, we might never discover the continent.
b0rsuk 6y ago
I wonder if someone comes up with a neat formula for the perimeter of an ellipse. Keppler gave up...
- pas 6y ago
  There are neat formulas, but those are just approximations, as an exact (closed form) is provably not possible. https://www.reddit.com/r/math/comments/8gw9on/is_it_possible_to_find_a_closed_formula_for_the/ https://www.reddit.com/r/math/comments/8gw9on/is_it_possible... https://www.youtube.com/watch?v=5nW3nJhBHL0 https://www.youtube.com/watch?v=5nW3nJhBHL0
alecbz 6y ago
I feel like "closed form" becomes a lot less special when you realize that things as simple as sin, cos, log, or even just sqrt don't have "closed forms" in the sense of "able to be expressed in terms of 'simpler' functions".
- analog31 6y ago
  Indeed, there's always a convention as to what the building blocks are. Like chemists don't look for how the quarks are configured -- they are satisfied to know how the atoms are arranged. In some cases, finding out what the building blocks are is an interesting problem in itself, for instance what things are computable with geometric construction.
  - ehvatum 6y ago
    > chemists don’t look for how the quarks are configured Neutron magnetic moment results from quark configuration, in terms of where the up and down quarks like to orbit each other within this gigantic neutron hadron thing. This is useful for understanding the magnetic properties of materials up to a hundred angstroms thick! Fire cold neutrons into your multi-angstrom Great Wall and see how they scatter.
- tikej 6y ago
  Oh it is special precisely because of that! We can almost always get the answer with numerical methods. Yet, numerical solutions are good almost only for certain values of parameters. Analytic solutions on the other hand provide more insight. It can be differentiated, integrated, compared and maybe (big maybe) transform to some other/more familiar form. Finally it can be connected to whole body of mathematical knowledge. Analytic solutions are very satisfying for that reasons. Closed solutions are small, very narrow subset of analytic ones, that’s why they feel special. But I suppose they become special only after some time spent with problems where closed form solutions are rare and unexpected. I suppose high school/college mathematics makes us way too used to rare special cases, such as closed form solutions.
z3t4 6y ago
Intuitively the formula should have the length of the rope, radius of the fence, and PI. And intuitively a rope length of the radius would cover half the circle, but a quick test of putting two goats in there shows they can't eat all the area. So the formula probably have a minus in it. Next I would try to figure out how large the goat circle outside the fence is because then i would also know the area inside...
kneedeepat 6y ago
As a Naval Academy grad, it still took me a second to figure out why it seemed to be a thing at the Naval Academy... But then Bill the Goat. FIN. Beat Army.
deepspace 6y ago
"[Fraser] worked out how long a rope is needed to allow a goat to graze in exactly half the volume of an n-dimensional sphere as n goes to infinity." That sounds way too much like the punchline of a stereotypical mathematician joke.
betwixthewires 6y ago
This is a very interesting problem. One thing I'd be interested in researching is the ratio between the 3 arcs in the scenario (https://en.m.wikipedia.org/wiki/File:Goat_problem_2D.svg https://en.m.wikipedia.org/wiki/File:Goat_problem_2D.svg), I'm sure this is a mathematical rabbit hole that would be exhausting to dive into.
tabtab 6y ago
Just use Goat Simulator: https://en.wikipedia.org/wiki/Goat_Simulator https://en.wikipedia.org/wiki/Goat_Simulator :-)
dartharva 6y ago
I'm not a science student so sorry if this sounds dumb, but can you not just do a computer simulation and run it (bruteforce) with varying rope lengths until you get the right one for one acre, then repeat the process for other areas till you get a list of areas and needed rope lengths, then do some regression on that list to get a close enough formula?
- phnofive 6y ago
  That’s essentially what this replaces - which had been done without the aid of simulations.
- dmurray 6y ago
  You just need to do the simulation once. It's an elementary result that if the solution for a circle of radius 1 is k, the solution for a circle of radius 2 will be 2k. That was already done in the 19th century without computers. But like you say, that gives only a "close enough" formula. Close enough for farmers, or even rocket engineers who might need much more precision, but not good enough for mathematicians.
amai 6y ago
Since when do we call a function of a ratio of two integrals a closed form solution? https://en.wikipedia.org/wiki/Goat_problem?Closed-form_solution https://en.wikipedia.org/wiki/Goat_problem?Closed-form_solut... https://en.wikipedia.org/wiki/Closed-form_expression?Comparison_of_different_classes_of_expressions https://en.wikipedia.org/wiki/Closed-form_expression?Compari...
- IshKebab 6y ago
  Yes I would describe that as an explicit equation rather than implicit.

Plaudits to Steve Nadis for pulling every goat joke known to man for this article.

> Unfortunately, there’s a catch. Ullisch’s solution ... can’t tell you, in a practical sense, how long to make the goat’s leash. Approximations are still required... Ok, then in what sense is this an exact solution?

whatshisface 6y ago
In the sense that sqrt(2) can be an exact solution, even though you can't write it down as a decimal.
- gowld 6y ago
  No, sqrt(2) is exact and no approximation is needed -- it's the diagnoal of a 1x1 square. Goats and grass don't care about decimal number systems. As the article says, the solution to the goat problem isn't practically constructible: "It’s a bit more abstruse — the ratio of two so-called contour integral expressions, with numerous trigonometric terms thrown into the mix"
  - whatshisface 6y ago
    >No, sqrt(2) is exact and no approximation is needed -- it's the diagonal of a 1x1 square. That's like saying that the goat-grass system as described is exact, and no approximation is needed. I can write 1 x sqrt(2) just as easily as I can write 1 x (goat-grass constant). We arbitrarily choose what constants and symbols are allowed when we use the phrase "exact solution." Philosophically, every solution is at most breaking down an answer into other solutions.
  - Koshkin 6y ago
    Problem is, if you make sqrt(2) exact - say, by an appropriate choice of the number system - then 1 will not be exact. This is because the ratio sqrt(2) / 1 is irrational in any number system.
    - thaumasiotes 6y ago
      But this isn't true. As gowld points out, you can easily represent an exact 1 and an exact sqrt(2) simultaneously; all you need is compass and straightedge. Interestingly, that's as far as it goes; if you start with a line of length 1, compass and straightedge will let you construct a line with length equal to the square root of any rational number. You can do addition, multiplication, subtraction, division... and square roots, and there things come to a stop.
      - whatshisface 6y ago
        But what singles out a compass and straightedge as a "the" construction system? There are infinity other ways to construct numbers, each one with a potentially different set of constructable numbers.
        thaumasiotes 6y ago
        You can use them to draw lines which you can then use to cut leather. They don't need to be "the" construction system; they are a construction system, and a very simple one.
        whatshisface 6y ago
        A pencil tied to the end of a string which is itself tied to the outside of a circle is also a construction system, one that happens to be able to construct the goat-grass area exactly. :)
        thaumasiotes 6y ago
        > one that happens to be able to construct the goat-grass area exactly. How? How did you get a string of the correct length? Note that the tool you described is a compass.
- deleted 6y ago
  [deleted]
castratikron 6y ago
It's in closed form
svat 6y ago
It's clearer from the Wikipedia article (https://en.wikipedia.org/w/index.php?title=Goat_problem&oldid=993368252 https://en.wikipedia.org/w/index.php?title=Goat_problem&oldi...): earlier we only had expressions for r like https://wikimedia.org/api/rest_v1/media/math/render/svg/9758becd77e7b4102ccdf50ff0845a749d236dc2 https://wikimedia.org/api/rest_v1/media/math/render/svg/9758... and https://wikimedia.org/api/rest_v1/media/math/render/svg/a63430b98b5e138a3d73831a2b7ecb6825be7a5b https://wikimedia.org/api/rest_v1/media/math/render/svg/a634... (from which of course one could numerically compute the answer r=1.1587…), but now thanks to this paper we have the expression https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7451ea92f107e56651328ad15bfcc3276f87f2 https://wikimedia.org/api/rest_v1/media/math/render/svg/5b74... that does not involve r on the right-hand side.
- Sniffnoy 6y ago
  Hm, if it's based on this ratio of contour integrals, shouldn't it be possible to do better than this? Like why would it be so hard to find the residues for these poles? Shouldn't that be just a bit of formal Laurent series manipulation? What am I missing here?
  - Someone 6y ago
    If the series don’t cancel out nicely (likely, I would guess), wouldn’t you end up with some infinite sum? If so, as this example shows (contour integrals in a closed form?) “closed form” is loosely defined, but I think most would say something with an infinite sum wouldn’t be one (but then, https://en.wikipedia.org/wiki/Closed-form_expression#Analytic_expression https://en.wikipedia.org/wiki/Closed-form_expression#Analyti... says: “In particular, special functions such as the Bessel functions and the gamma function are usually allowed, and often so are infinite series and continued fractions. On the other hand, limits in general, and integrals in particular, are typically excluded.[citation needed]”
    - Sniffnoy 6y ago
      No, you shouldn't end up with an infinite sum; if you only want to know the series finitely far out, you only need to know finitely many terms. However, it seems the problem is hard for other reasons. I had assumed, without checking, that they'd put the center of the circle at 3pi/8 because it's a pole. Nope! As best I can tell from some graphing tools, there is indeed precisely one pole in the circle, and it's on the real line, but it's close to the right endpoint; I don't know that it has any nice form. So I imagine that getting any sort of exact series expansion around there -- or even just getting exactly the first few terms, i.e. the first few derivatives there, which would be all you'd need -- would be difficult for that reason. (Although, the higher the order of the zero, the more initial terms you'd need...) Edit: Actually, I guess it looks like a zero of order 1? Except that doesn't make sense, because then the top contour integral would be zero...
6gvONxR4sf7o 6y ago
Imagine an exact solution like this: sin(cos(log(98234/123)+tan(exp(pi/5))-1)+pi + integral(complicated function(x) dx from 0 to 1) It doesn't tell you in a practical sense how long to make the goat's leash. So you find the numerical version to however many decimals you want to get a useful approximation.
pfortuny 6y ago
The unknown is isolated: r=very-difficult-to-compute-expression
elil17 6y ago
The word exact here is an inexact description of what sort of solution this really is - it's a closed form explicit solution. A closed form solution means that the equation is limited to to certain common mathematical operations and is finite in length. An explicit solution means that the quantity we are solving for is isolated on one side of the equation.

2sk21 6y ago

Is this in any way to the problem of quadrature of the lune? https://en.wikipedia.org/wiki/Lune_of_Hippocrates https://en.wikipedia.org/wiki/Lune_of_Hippocrates

svat 6y ago

The paper this article is about: * A Closed-Form Solution to the Geometric Goat Problem, by Ingo Ullisch in The Mathematical Intelligencer. https://doi.org/10.1007/s00283-020-09966-0 https://doi.org/10.1007/s00283-020-09966-0 The illustration in the article (I mean the top half of https://d2r55xnwy6nx47.cloudfront.net/uploads/2020/12/Grazing-goat-figure.svg https://d2r55xnwy6nx47.cloudfront.net/uploads/2020/12/Grazin...) was really helpful to understand what the question is. I know Quanta Magazine articles sometimes get some negative comments here on HN from readers who expect a different kind of exposition than what is suitable for a magazine of that sort, but for my part I'm really grateful to Quanta Magazine for bringing attention to papers like this, and writing nice articles about them with history, good illustrations, quotes from other mathematicians, etc. For the impatient, Wikipedia has a short article on the problem and earlier progress: https://en.wikipedia.org/w/index.php?title=Goat_problem&oldid=993368252 https://en.wikipedia.org/w/index.php?title=Goat_problem&oldi... From this it's more clear in what sense this is progress: whereas earlier the answer r=1.1587284730181215178... (https://oeis.org/A133731 https://oeis.org/A133731) was known via more than one formula of the form r = (some function of r), only now do we have a closed-form expression of the form r = (some expression not involving r).

tunesmith 6y ago

I first misread "goat tied to the inside of the fence" as meaning tied to a post in the center of the circle, and was confused at why everyone thought this question was so hard.

mrlala 6y ago
I was just drawing what you said on the whiteboard, and I was like.. uh, am I missing something or am I a super genius? I am positive the latter is incorrect. I had to re-read the first paragraph multiple times until I understood the goat was not tethered to the center.
- InitialLastName 6y ago
  There's a picture of the problem like halfway down the article.
  - mrlala 6y ago
    Yeah but I wanted to attempt to think about the problem myself before reading more on the article in case they gave away a very simple clue.
fastaguy88 6y ago
I initially assumed that the goat was tied to the fence in such a way that the non-goat rope-end could move completely around the circumference of the fence and not be fixed to a single point.
- acqq 6y ago
  I was also at first confused, and I find the problem as published in 1894 (i.e. almost 130 years ago) much clearer: "A circle containing one acre is cut by another whose centre is on the circumference of the given circle, and the area common to both is one-half acre. Find that radius of the cutting circle." I think that formulation had to be at least mentioned near the beginning of the text, when not used in the first sentence.

zaptheimpaler 6y ago

Finally! This was getting to be a real problem in my grazing goat as a service business.

hinkley 6y ago
Goats as a Service has been around longer than The Cloud. I think you already missed that gold rush.
_jal 6y ago
You laugh, but growing up, my family used goats to keep brush down on our property, and would occasionally loan them to neighbors for the same.
- mhh__ 6y ago
  https://www.telegraph.co.uk/technology/google/5297097/Google-hires-goats-to-cut-grass.html https://www.telegraph.co.uk/technology/google/5297097/Google...
skinkestek 6y ago
I think that is an actual service provided in Norway.
- codetrotter 6y ago
  You are correct. I remember seeing some goats at Oscarsborg Fortress and I was told that they were there on loan or on hire. Looked it up now and it checks out. The articles in the two links below affectionately refer to the featured goats as the "coast goat commando", which I think is just lovely :) https://www.oblad.no/badebyen/kystgeitkommandoen-til-oscarsborg/s/2-2.2610-1.3865150 https://www.oblad.no/badebyen/kystgeitkommandoen-til-oscarsb... https://www.oblad.no/badebyen/umb-geitene-pa-sommerjobb/s/2-2.2610-1.3884117 https://www.oblad.no/badebyen/umb-geitene-pa-sommerjobb/s/2-... Both of these articles are in Norwegian but basically they talk about the importance of keeping the vegetation in check, and that the goats are great at this, as well as the social value that goats provide to visitors. The goats featured here are owned by a University and were rented out to the Oscarsborg Fortress. There are pictures of the goats also in the articles. PS: For anyone not familiar with Oscarsborg check out the following links for some info and pictures: https://en.wikipedia.org/wiki/Oscarsborg_Fortress https://en.wikipedia.org/wiki/Oscarsborg_Fortress https://www.forsvarsbygg.no/no/festningene/finn-din-festning/oscarsborg-festning/ https://www.forsvarsbygg.no/no/festningene/finn-din-festning... https://en.wikipedia.org/wiki/Battle_of_Drøbak_Sound https://en.wikipedia.org/wiki/Battle_of_Drøbak_Sound Using the defence batteries at Oscarsborg Fortress, the Norwegian coastal defence successfully sank the German heavy cruiser Blücher on 9 April 1940, forcing the German fleet to fall back. https://en.wikipedia.org/wiki/German_cruiser_Blücher https://en.wikipedia.org/wiki/German_cruiser_Blücher

atsmyles 6y ago

Hey Terence Tao, you might be a great mathematician but Ingo Ullisch is the GOAT!

deleted 6y ago

[deleted]

nullc 6y ago

Why do I have to go to Wikipedia to see the solution? Would it have been that hard to stick https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7451ea92f107e56651328ad15bfcc3276f87f2 https://wikimedia.org/api/rest_v1/media/math/render/svg/5b74... in the article?

ABeeSea 6y ago
From a journalistic standpoint, including the formula would require a longer digression into contour integrals than the one clause the article currently contains. It’s also not clear what license that image is available under.
- segfaultbuserr 6y ago
  > It’s also not clear what license that image is available under. A plain image of mathematical equation is not copyrightable, it's literally the MathJax output of a LaTeX equation ({\displaystyle r=2\cos \left({\frac {1}{2}}{\frac {\oint _{|z-3\pi /8|=\pi /4}z/(\sin z-z\cos z-\pi /2)\,dz}{\oint _{|z-3\pi /8|=\pi /4}1/(\sin z-z\cos z-\pi /2)\,dz}}\right)}). It does not have any copyrightable artistic design. And to the nitpickers - the pixmap output of a font is also not copyrightable under U.S. copyright laws. Even if it is, Computer Modern is available under a free license. And even if it's not, pure facts - such as math formulas - are not copyrightable, it would be trivial to write down the identical equation using another program and font. Copyright is not an inevitable, divine, or natural right, it is only an utilitarian tool adopted by the Constitution and lawmakers "to promote the Progress of Science and useful Arts." Thus, there always exist things that cannot be copyrighted. It's also why fair use of copyrighted works is conditionally allowed (not relevant to this case). The tendency of people to assume that every single piece of data is automatically controlled exclusively under copyright is frustrating.
- blu_ 6y ago
  > Content is available under CC BY-SA 3.0 unless otherwise noted Just including the actual solution in an article about this problem would be IMO a much better choice.
6y ago

crazygringo 6y ago

It's fascinating how a problem that looks at first glance like it belongs as a basic homework problem in a calculus textbook turns out to be so difficult. I'm curious if there are lists of other problems that are similarly easy to understand in a few seconds, that seem like they'd be similarly easy to solve, but which turn out to be fiendishly hard like this one? Especially ones that can be visualized easily geometrically like this one.

darkhorse13 6y ago
>I'm curious if there are lists of other problems that are similarly easy to understand in a few seconds I too would love such a list. Can anyone point me if one does exist?
- deleted 6y ago
  [deleted]
FartyMcFarter 6y ago
I'm struggling to come up with examples. Maybe the 3n+1 problem? It's definitely easy to understand in a few seconds, and definitely fiendishly hard. I'm not sure if it seems easy to solve though :) Another one that came to mind is about efficiently computing the permanent of a matrix[1]. Maybe understandable in seconds if you know how determinants work. [1] https://en.wikipedia.org/wiki/Computing_the_permanent https://en.wikipedia.org/wiki/Computing_the_permanent
- DataDaoDe 6y ago
  3n + 1 is interesting because it seems so tantalizingly easy and beautiful at first glance. Its like looking at a stream where you can clearly see the stony bed and it appears to be ankle deep but once you take a step you realize the lighting has fooled you and you fall head over heels into icy waters.
  - pas 6y ago
    It's so much more interesting if you happen to know about the "hydra game" (the Goodstein theorem), which is also a question about sequences and their termination at 0. https://en.wikipedia.org/wiki/Goodstein%27s_theorem https://en.wikipedia.org/wiki/Goodstein%27s_theorem But this is easily proved (using infinite ordinals, but it seems it could be proved just by coming up with a similar concept like the infinite ordinal arithmetic, basically providing an upper bound for each step of the algorithm and showing that there's an eventual maximum to these and then there's a monotonic decrease, and that the number of steps are always finite).

r-bryan 6y ago

Cool. Now solve it for a circular fence on the surface of a sphere. In fact, solve all four cases {exterior, interior of sphere} X {exterior, interior of fence}. Spherical trig can only make the solution(s) even more heroic, right?

Akronymus 6y ago
Why not try hyperbolic space too?
antiquark 6y ago
You forgot about the hypergoats.
scatters 6y ago
Surface of a sphere depends on how large the fence is compared to the sphere. If the fence is small, the answer is the same as the plane version. If the fence is as large as possible (an equator) then the rope needs to be precisely one quadrant in length, equalling the "radius" of the fence. If the fence encloses more than half the sphere... well, if it encloses all of the sphere (that is, it is a small circle with the "inside" declared to be the outside) then the rope is again one quadrant, so half the "radius" of the fence. More interesting is a space-goat tethered to the interior of a hollow sphere, hypersphere etc.; no closed-form solutions for higher-dimensional cases, but the answer tends to sqrt(2) as the number of dimensions approaches infinity.

cochne 6y ago

Not really an exact or closed form solution since it contains an integral. It is an explicit solution compared to the implicit one we had before.

adwn 6y ago
> Not really an exact [...] solution But it is exact. > Not really [a] closed form solution It's my understanding that what is and isn't "closed form" is rather arbitrary. Functions which are used frequently – like exp() – are elevated to closed form status, and yet you can't evaluate exp() in a finite number of steps. So how is the explicit solution to the goat problem objectively different?
- cochne 6y ago
  Yes I also feel that way to some degree, but I just never considered an integral to be closed form. There is some argument to be made for exp, as it is considered an 'elementary function'. I was going off this statement from wikipedia on closed form expressions "It may contain constants, variables, certain "well-known" operations (e.g., + − × ÷), and functions (e.g., nth root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit, differentiation, or integration." Edit: I think if you say an integral is closed form, you must also admit that a limit is closed form, since an integral is defined in terms of limits (though technically more restrictive). In that case, you should also admit that we already had a closed form expression for this number, as it could be expressed as a limit of an iterative process.
  - adwn 6y ago
    > There is some argument to be made for exp, as it is considered an 'elementary function'. But exp() is defined as the limit over an infinite sum, so why does it get to be an elementary function? My point it that the distinction between closed form and non-closed form is arbitrary, and that there is no qualitative difference. In fact, limit and integral are just (higher-order) functions as well – and rather ubiquitous ones, so why aren't they considered elemental?

skinkestek 6y ago

This is probably written by people who don't have real-life experience with goats I guess. I cannot even concentrate on the problem since I keep coming back to what the goat will do: - jump the fence - chew the rope - probably more

yetihehe 6y ago
In such cases, it's best to assume it's a special zero dimensional point goat, which only chews grass.
- adwn 6y ago
  ...in a vacuum.
- cgriswald 6y ago
  How do you tie a zero dimensional point goat? How do you know he's in the fence at all? How does he chew the grass? Where does the grass go?
  - tuatoru 6y ago
    And having discovered the goat, how do you put a tether on something with no size?
  - WJW 6y ago
    You'd need a special rope to tie it as well, in addition to low-dimensional fencing and specialized grass.
  - thechao 6y ago
    Literally every one of these questions pertains to real goats, too.
  - morei 6y ago
    ... are you serious? You use an infinitely thin rope of course! Slightly more seriously, you take the limit of a alpha-sized goat tied with an alpha-sized rope as alpha goes to zero from above.

tuatoru 6y ago

From TFA: * Of course, it won’t upend textbooks or revolutionize math research, Ullisch concedes, because this problem is an isolated one. “It’s not connected to other problems or embedded within a mathematical theory.”* A island peak hinting at a submerged continent of mathematics. Unfortunately since our brains evolved (under a regime of calorie cost vs survival benefit) and are therefore limited, we might never discover the continent.

b0rsuk 6y ago

I wonder if someone comes up with a neat formula for the perimeter of an ellipse. Keppler gave up...

pas 6y ago
There are neat formulas, but those are just approximations, as an exact (closed form) is provably not possible. https://www.reddit.com/r/math/comments/8gw9on/is_it_possible_to_find_a_closed_formula_for_the/ https://www.reddit.com/r/math/comments/8gw9on/is_it_possible... https://www.youtube.com/watch?v=5nW3nJhBHL0 https://www.youtube.com/watch?v=5nW3nJhBHL0

alecbz 6y ago

I feel like "closed form" becomes a lot less special when you realize that things as simple as sin, cos, log, or even just sqrt don't have "closed forms" in the sense of "able to be expressed in terms of 'simpler' functions".

analog31 6y ago
Indeed, there's always a convention as to what the building blocks are. Like chemists don't look for how the quarks are configured -- they are satisfied to know how the atoms are arranged. In some cases, finding out what the building blocks are is an interesting problem in itself, for instance what things are computable with geometric construction.
- ehvatum 6y ago
  > chemists don’t look for how the quarks are configured Neutron magnetic moment results from quark configuration, in terms of where the up and down quarks like to orbit each other within this gigantic neutron hadron thing. This is useful for understanding the magnetic properties of materials up to a hundred angstroms thick! Fire cold neutrons into your multi-angstrom Great Wall and see how they scatter.
tikej 6y ago
Oh it is special precisely because of that! We can almost always get the answer with numerical methods. Yet, numerical solutions are good almost only for certain values of parameters. Analytic solutions on the other hand provide more insight. It can be differentiated, integrated, compared and maybe (big maybe) transform to some other/more familiar form. Finally it can be connected to whole body of mathematical knowledge. Analytic solutions are very satisfying for that reasons. Closed solutions are small, very narrow subset of analytic ones, that’s why they feel special. But I suppose they become special only after some time spent with problems where closed form solutions are rare and unexpected. I suppose high school/college mathematics makes us way too used to rare special cases, such as closed form solutions.

z3t4 6y ago

Intuitively the formula should have the length of the rope, radius of the fence, and PI. And intuitively a rope length of the radius would cover half the circle, but a quick test of putting two goats in there shows they can't eat all the area. So the formula probably have a minus in it. Next I would try to figure out how large the goat circle outside the fence is because then i would also know the area inside...

kneedeepat 6y ago

As a Naval Academy grad, it still took me a second to figure out why it seemed to be a thing at the Naval Academy... But then Bill the Goat. FIN. Beat Army.

deepspace 6y ago

"[Fraser] worked out how long a rope is needed to allow a goat to graze in exactly half the volume of an n-dimensional sphere as n goes to infinity." That sounds way too much like the punchline of a stereotypical mathematician joke.

betwixthewires 6y ago

This is a very interesting problem. One thing I'd be interested in researching is the ratio between the 3 arcs in the scenario (https://en.m.wikipedia.org/wiki/File:Goat_problem_2D.svg https://en.m.wikipedia.org/wiki/File:Goat_problem_2D.svg), I'm sure this is a mathematical rabbit hole that would be exhausting to dive into.

tabtab 6y ago

Just use Goat Simulator: https://en.wikipedia.org/wiki/Goat_Simulator https://en.wikipedia.org/wiki/Goat_Simulator :-)

dartharva 6y ago

I'm not a science student so sorry if this sounds dumb, but can you not just do a computer simulation and run it (bruteforce) with varying rope lengths until you get the right one for one acre, then repeat the process for other areas till you get a list of areas and needed rope lengths, then do some regression on that list to get a close enough formula?

phnofive 6y ago
That’s essentially what this replaces - which had been done without the aid of simulations.
dmurray 6y ago
You just need to do the simulation once. It's an elementary result that if the solution for a circle of radius 1 is k, the solution for a circle of radius 2 will be 2k. That was already done in the 19th century without computers. But like you say, that gives only a "close enough" formula. Close enough for farmers, or even rocket engineers who might need much more precision, but not good enough for mathematicians.

amai 6y ago

Since when do we call a function of a ratio of two integrals a closed form solution? https://en.wikipedia.org/wiki/Goat_problem?Closed-form_solution https://en.wikipedia.org/wiki/Goat_problem?Closed-form_solut... https://en.wikipedia.org/wiki/Closed-form_expression?Comparison_of_different_classes_of_expressions https://en.wikipedia.org/wiki/Closed-form_expression?Compari...

IshKebab 6y ago
Yes I would describe that as an explicit equation rather than implicit.